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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 22, 2015

Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

  • Emile Franc Doungmo Goufo and Stella Mugisha
From the journal Open Mathematics

Abstract

Classical models of clusters’ fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order γ (with 0 < γ ≤ 1). In this article, we introduce and study a model that can be understood as the fractional generalization of the clusters’ fission process.We make use of the theory of strongly continuous solution operators for fractional models (analogues of C0-semigroups for classical models) and the subordination principle for fractional evolution equations (Bazhlekova, 2000, Prüss, 1993) to analyze and show existence results for clusters’ splitting model with derivative of fractional order. In the process, we exploit some properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005), the He’s homotopy perturbation (He, 1999) and Kato’s type perturbation (Banasiak, 2006) methods. The Cauchy problem for multiplication operator in the fractional dynamics is first considered, before we perturb it. Some additional concepts like Laplace transform, Hille-Yosida theorem and the dominated convergence theorem are use to finally show that there is a solution operator to the full fractional model that is positive and contractive.

References

[1] Anderson W.J., Continuous-Time Markov Chains. An Applications-Oriented Approach, Springer Verlag, New York, 1991 10.1007/978-1-4612-3038-0Search in Google Scholar

[2] Atangana A., On the singular perturbations for fractional differential equation, The Scientific World Journal, 2014, Article ID 752371, vol. 2014, 9 pages, preprint available at http://dx.doi.org/10.1155/2013/752371 Search in Google Scholar

[3] Atangana A., Kílíçman A., A possible generalization of acoustic wave equation using the concept of perturbed, Mathematical problems in Engineering, Article ID 696597 preprint available at http://dx.doi.org/10.1155/2013/696597 10.1155/2013/696597Search in Google Scholar

[4] Atangana A., Botha F.C., A generalized groundwater flow equation using the concept of variable-order derivative, Boundary Value Problems 2013, 2013:53 preprint available at http://www.boundaryvalueproblems.com/content/2013/1/53 10.1186/1687-2770-2013-53Search in Google Scholar

[5] Balakrishnan, A.V., Fractional powers of closed operators and semigroups generated by them, Pacific J. Math., 1960, 10, 419 10.2140/pjm.1960.10.419Search in Google Scholar

[6] Banasiak J., Arlotti L., Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, 2006. Search in Google Scholar

[7] Bartle R.G., The elements of integration and Lebesgue measure, Wiley Interscience, 1995 10.1002/9781118164471Search in Google Scholar

[8] Benson D.A., Schumer R., Meerschaert M.M., Wheatcraft S.W., Fractional Dispersion, Levy Motion, and the MADE Tracer Tests. Transport in Porous Media 2001, 42: 211-240. 10.1007/978-94-017-1278-1_11Search in Google Scholar

[9] Benson D.A., Meerschaert M.M., Revielle J., Fractional calculus in hydrologic modeling: A numerical perspective. Advances in Water Resources 2013, 51, 479–497 10.1016/j.advwatres.2012.04.005Search in Google Scholar PubMed PubMed Central

[10] Bazhlekova E. G., Subordination principle for fractional evolution equations, Fractional Calculus & Applied Analysis, 2000, 3(3), 213 – 230 Search in Google Scholar

[11] Berberan-Santos Mário N., Properties of the Mittag-Leffler relaxation function, Journal of Mathematical Chemistry, November 2005, 38(4) 10.1007/s10910-005-6909-zSearch in Google Scholar

[12] Brockmann D., Hufnagel L., Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Phys. Review Lett. 98, 2007 10.1103/PhysRevLett.98.178301Search in Google Scholar

[13] Caputo M., Linear models of dissipation whose Q is almost frequency independent, Journal of the Royal Australian Historical Society, 1967, 13(2), 529–539 10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[14] Diethelm K., The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. 10.1007/978-3-642-14574-2Search in Google Scholar

[15] Demirci E., Unal A., Özalp N., A fractional order seir model with density dependent death rate, Hacettepe Journal of Mathematics and Statistics, 2011, 40(2), 287–295 Search in Google Scholar

[16] Doungmo Goufo E.F., Maritz R. , Munganga J., Some properties of Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence, Advances in Difference Equations 2014, 2014:278. preprint available at DOI: 10.1186/1687-1847-2014- 278, URL: http://www.advancesindifferenceequations.com/content/2014/1/278 10.1186/1687-1847-2014-278Search in Google Scholar

[17] Doungmo Goufo E.F., Mugisha S., Mathematical solvability of a Caputo fractional polymer degradation model using further generalized functions, Mathematical Problems in Engineering, Volume 2014, Article ID 392792, 5 pages, 2014. preprint available at http://dx.doi.org/10.1155/2014/392792 10.1155/2014/392792Search in Google Scholar

[18] EF Doungmo Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fract. Calc. Appl. Anal., 18(3), 2015 10.1515/fca-2015-0034Search in Google Scholar

[19] EF Doungmo Goufo, Non-local and Non-autonomous Fragmentation-Coagulation Processes in Moving Media, PhD thesis, North- West University, South Africa, 2014. Search in Google Scholar

[20] Doungmo Goufo E.F., Oukouomi Noutchie S.C., Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates, Comptes Rendus Mathematique, C.R Acad. Sci, Paris, Ser, I, 2013, preprint available at http://dx.doi.org/10.1016/j.crma.2013.09.023 10.1016/j.crma.2013.09.023Search in Google Scholar

[21] Engel K-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194), Springer, 2000 Search in Google Scholar

[22] Érdelyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, Vol. III McGraw-Hill, New York, 1955 Search in Google Scholar

[23] Filippov I., On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl. 1961, 6, 275–293 10.1137/1106036Search in Google Scholar

[24] Garibotti C. R., Spiga G., Boltzmann equation for inelastic scattering, J. Phys. A 1994, 27, 2709–2717 10.1088/0305-4470/27/8/009Search in Google Scholar

[25] Gel’fand I., Shilov G., Generalized Functions, vol. I. Academic Pres5, New York, 1964. Search in Google Scholar

[26] Gorenflo R., Luchko Y., Mainardi F., Analytical properties and applications of the Wright function, Fractional Calculus and Applied Analysis, 1999, 2(4), 383–414 Search in Google Scholar

[27] He J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech., 1999 vol. 178, pp. 257–262 10.1016/S0045-7825(99)00018-3Search in Google Scholar

[28] He J. H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Internat. J. Non-Linear Mech., 2000, 35, 37–43 10.1016/S0020-7462(98)00085-7Search in Google Scholar

[29] Hilfer R., Application of Fractional Calculus in Physics, World Scientific, Singapore, 1999. 10.1142/3779Search in Google Scholar

[30] Hilfer R., On new class of phase transitions, Random magnetism and High temprature superconductivity, page 85, Singapore, World Scientific publ. Co., 1994. Search in Google Scholar

[31] Lachowicz M., Wrzosek D., A nonlocal coagulation-fragmentation model, Appl. Math. (Warsaw), 2000, 27 (1), 45–66 10.4064/am-27-1-45-66Search in Google Scholar

[32] Lions J.L., Peetre J., Sur une classe d’espace d’interpolation, Inst. Hautes étude Sci. Publ. Math, 1964, 19, 5–68 10.1007/BF02684796Search in Google Scholar

[33] McLaughlin D. J., Lamb W., McBride A. C., A semigroup approach to fragmentation models, SIAM Journal on Mathematical Analysis, 1997, 28(5), 1158–1172 10.1137/S0036141095291701Search in Google Scholar

[34] Majorana A., Milazzo C., Space homogeneous solutions of the linear semiconductor Boltzmann equation. J. Math. Anal. Appl. 2001, 259(2), 609–629 10.1006/jmaa.2001.7444Search in Google Scholar

[35] Melzak Z.A., A Scalar Transport Equation, Trans. Amer. Math. Soc., 1957, 85, 547–560 10.1090/S0002-9947-1957-0087880-6Search in Google Scholar

[36] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Willey and Sons, Inc., New York, 2003. Search in Google Scholar

[37] Mittag-Leffler G.M., C. R. Acad. Sci. Paris (Ser. II) 1903, 137–554 Search in Google Scholar

[38] Norris J.R., Markov Chains, Cambridge University Press, Cambridge, 1998 Search in Google Scholar

[39] Oldham K. B., Spanier J., The fractional calculus, Academic Press, New York, 1999 Search in Google Scholar

[40] Oukouomi Noutchie S.C., Doungmo Goufo E. F., Exact solutions of fragmentation equations with general fragmentation rates and separable particles distribution kernels, Mathematical Problems in Engineering, 2014, vol. 2014, Article ID 789769, 5 pages, preprint available at http://dx.doi.org/10.1155/2014/789769 10.1155/2014/789769Search in Google Scholar

[41] Oukouomi Noutchie S.C., Doungmo Goufo E.F., Global solvability of a continuous model for nonlocal fragmentation dynamics in a moving medium, Mathematical Problem in Engineering, 2013, vol. 2013, Article ID 320750, 8 pages, 2013, preprint available at http://dx.doi.org/10.1155/2013/320750 10.1155/2013/320750Search in Google Scholar

[42] Özalp N., Demirci E., A fractional order SEIR model with vertical transmission Mathematical and Computer Modelling, 54(2011), 1–6 10.1016/j.mcm.2010.12.051Search in Google Scholar

[43] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, 44, 1983 10.1007/978-1-4612-5561-1Search in Google Scholar

[44] Podlubny, I., Fractional Differential Equations, Academic Press, California, USA, 1999 Search in Google Scholar

[45] Pooseh S., Rodrigues H.S., Torres D.F.M., Fractional derivatives in dengue epidemics. In: Simos, T.E., Psihoyios, G., Tsitouras, C., Anastassi, Z. (eds.) Numerical Analysis and Applied Mathematics, ICNAAM, American Institute of Physics, Melville, 2011, 739–742 10.1063/1.3636838Search in Google Scholar

[46] Prüss J., Evolutionary Integral Equations and Applications, Birkhäuser, Basel–Boston–Berlin 1993 10.1007/978-3-0348-8570-6Search in Google Scholar

[47] Rudnicki R., Wieczorek R., Phytoplankton dynamics: From the behaviour of cells to a transport equation, Math. Model. Nat. Phenom, 2006, 1 (1), 83–100 10.1051/mmnp:2006005Search in Google Scholar

[48] Rubin B., Fractional Integrals and potentials, Addison Wesley Longman Limited, Harlow 1996 Search in Google Scholar

[49] Samko S.G., Kilbas A.A., Marichev O.I., Franctional integrals and derivatives, Theory and Application, Gordon and Breach, Amsterdam, 1993 Search in Google Scholar

[50] Wagner W., Explosion phenomena in stochastic coagulation-fragmentation models, Ann. Appl. Probab., 2005, 15(3), 2081–2112 10.1214/105051605000000386Search in Google Scholar

[51] Westphal U., ein Kalkül für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren, Teil I: Halbgruppen-erzeuger, Compositio Math., 1970, 22, 67–103 Search in Google Scholar

[52] Wright E. M., The generalized Bessel function of order greater than one. Quarterly Journal of Mathematics (Oxford ser.), 1940, 11, 36–48 10.1093/qmath/os-11.1.36Search in Google Scholar

[53] Yosida K., Functional Analysis, Sixth Edition, Springer- Verlag, 1980 Search in Google Scholar

[54] Ziff R.M., McGrady E.D., The kinetics of cluster fragmentation and depolymerization, J. Phys. A, 1985, 18 3027–3037 10.1088/0305-4470/18/15/026Search in Google Scholar

[55] Ziff R.M., McGrady E.D., Shattering transition in fragmentation, Phys. Rev. Lett., 1987, 58(9) 10.1103/PhysRevLett.58.892Search in Google Scholar PubMed

[56] Ziff, R.M., McGrady, E.D., Kinetics of polymer degradation, Macromolecules 19, 1986, 2513–2519. 10.1021/ma00164a010Search in Google Scholar

Received: 2014-4-15
Accepted: 2014-12-24
Published Online: 2015-5-22

©2015 Emile Franc Doungmo Goufo and Stella Mugisha

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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